The Riemann Hypothesis for Polynomials Orthogonal on the Unit Circle

Abstract

AbstractAn axiomatic treatment is first given of certain Hilbert spaces of polynomials which are implicit in the Szegö theory of polynomials orthogonal on the unit circle [7]. Methods of factorization theory are applied to determine the isometric inclusion of such spaces. Every Szegö space is determined by a polynomial which has no zeros in the unit disk and by an integer r which is greater than or equal to the degree of the polynomial. A positivity condition is formulated for a Szegö space in terms of a quantum q, which is a given number with 0 < q < 1. The quantum can be used to state a functional identity for the defining polynomial. The quantum positivity condition implies that the zeros of the defining polynomial lie on a circle determined by the functional identity. Even when no functional identity is satisfied, the quantum positivity condition denies the existence of twin zeros which are symmetrically placed about the circle. The quantum positivity condition is also applied to weighted Hardy spaces. A characterization is given of those weight functions for which the associated space satisfies the quantum positivity condition. A theorem on the asymptotic behavior of derived Szegö spaces is obtained under a quantum positivity hypothesis for weighted Hardy space. These results are used in an interpretation of the convergence of the Euler product for the zeta‐function of a function field with finite constants field.